Examples
- The identity map and zero map are linear.
- The map, where c is a constant, is linear.
- For real numbers, the map is not linear.
- For real numbers, the map is not linear (but is an affine transformation, and also a linear function, as defined in analytic geometry.)
- If A is a real m × n matrix, then A defines a linear map from Rn to Rm by sending the column vector x ∈ Rn to the column vector Ax ∈ Rm. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.
- The (definite) integral is a linear map from the space of all real-valued integrable functions on some interval to R
- The (indefinite) integral (or antiderivative) is not considered a linear transformation, as the use of a constant of integration results in an infinite number of outputs per input.
- Differentiation is a linear map from the space of all differentiable functions to the space of all functions.
- If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dimF(W) × dimF(V) matrices in the way described in the sequel are themselves linear maps.
- The expected value of a random variable is linear, as for random variables X and Y we have E = E + E and E = aE, but the variance of a random variable is not linear, as it violates the second condition, homogeneity of degree 1: V = a2V.
Read more about this topic: Linear Map
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